The Decline Types

he basic parameter of conventional decline curve analysis is the fractional decline rate, D, defined as the change in production rate per unit time per unit production rate. Mathematically, D is defined by

D = -(1/q)*(dq/dt)

where q is the production rate, t is the time, and dq/dt is the derivative of production rate with respect to time. The dimension of D is the reciprocal of time.

After studying the production rate history from 149 fields, Arps proposed three equations to describe the production rate decline in petroleum reservoirs: the exponential; the hyperbolic; and the harmonic equation. The three decline equations differ by the way the fractional decline rate, D, is assumed to vary with time and production rate. The exponential decline equation assumes D is constant. As a result, it is commonly referred to as the constant decline equation. The harmonic decline equation assumes D is proportional to the production rate. The hyperbolic decline equation assumes D is proportional to the production rate raised to the power n, where n is a fraction between zero and one.

Inherent Assumptions

Arps proposed three decline equations on purely empirical grounds and without any theoretical basis. However, even through the reservoirs used by Arps covered a wide rage of reservoir characteristics and drive mechanisms, they shared a common operating mode, spacing, and productivity range. This imposed some inherent assumptions:

1. The production rate data was in the boundary dominated flow regime. This is equivalent to the pseudo steady state flow regime observed under constant rate production in pressure transient testing. The time to reach this flow regime depends on the permeability, the porosity, the total system compressibility, the fluid viscosity, and the drainage area. The reservoirs studied by Arps had relatively high permeability and were developed using closer well spacing. As a result, the data Arps used mainly in the boundary dominated flow regime. These equations should not be used, for example, in tight gas reservoirs, where the time to reach the boundary dominated flow may take months to years.

2. The flowing bottom hole pressure is constant. The reservoirs considered by Arps were operated without any allowable restrictions, and were produced wide open near maximum production rate. This is close to a constant flowing bottom hole pressure production.

3. The well is continuously on production. In practice, this is usually not the case, since a well is frequently shut in for various operational reasons. The effects of any production interruption can be removed by plotting the data in terms of longer time units, such as every month, six months, or a year.

The Decline Exponent and the Drive Mechanism

We know that higher exponents indicate a slower decline and higher recovery. Therefore, we expect a water drive reservoir, for example, to exhibit a higher exponent than a solution gas drive.

The expected range of the exponent, based on Arps data, is between zero and 0.5. Values higher than one are usually the result of using conventional decline analysis to production rate data in the transient flow regime.

WOR versus Cumulative Produced Oil

A common method of analyzing the performance of waterfloods is to plot the WOR versus cumulative produced oil for a field or group of wells. In mature (post-break through) waterfloods the log of WOR versus cumulative produced oil is a linear relationship. Oil rate decline curves obtained from this method are hyperbolic (or harmonic) in nature.

This technique is based on the one-dimensional Buckley-Leverett frontal advance equation for linear, immiscible displacement.

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